L(s) = 1 | + (0.377 + 0.926i)2-s + (0.775 + 0.631i)3-s + (−0.715 + 0.699i)4-s + (0.877 − 0.480i)5-s + (−0.291 + 0.956i)6-s + (−0.538 − 0.842i)7-s + (−0.917 − 0.398i)8-s + (0.203 + 0.979i)9-s + (0.775 + 0.631i)10-s + (0.203 + 0.979i)11-s + (−0.995 + 0.0909i)12-s + (−0.746 − 0.665i)13-s + (0.576 − 0.816i)14-s + (0.983 + 0.181i)15-s + (0.0227 − 0.999i)16-s + (−0.334 + 0.942i)17-s + ⋯ |
L(s) = 1 | + (0.377 + 0.926i)2-s + (0.775 + 0.631i)3-s + (−0.715 + 0.699i)4-s + (0.877 − 0.480i)5-s + (−0.291 + 0.956i)6-s + (−0.538 − 0.842i)7-s + (−0.917 − 0.398i)8-s + (0.203 + 0.979i)9-s + (0.775 + 0.631i)10-s + (0.203 + 0.979i)11-s + (−0.995 + 0.0909i)12-s + (−0.746 − 0.665i)13-s + (0.576 − 0.816i)14-s + (0.983 + 0.181i)15-s + (0.0227 − 0.999i)16-s + (−0.334 + 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.037429777 - 0.4597792444i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.037429777 - 0.4597792444i\) |
\(L(1)\) |
\(\approx\) |
\(1.157210333 + 0.6452572812i\) |
\(L(1)\) |
\(\approx\) |
\(1.157210333 + 0.6452572812i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.377 + 0.926i)T \) |
| 3 | \( 1 + (0.775 + 0.631i)T \) |
| 5 | \( 1 + (0.877 - 0.480i)T \) |
| 7 | \( 1 + (-0.538 - 0.842i)T \) |
| 11 | \( 1 + (0.203 + 0.979i)T \) |
| 13 | \( 1 + (-0.746 - 0.665i)T \) |
| 17 | \( 1 + (-0.334 + 0.942i)T \) |
| 19 | \( 1 + (0.158 - 0.987i)T \) |
| 23 | \( 1 + (-0.962 - 0.269i)T \) |
| 29 | \( 1 + (-0.203 + 0.979i)T \) |
| 31 | \( 1 + (0.803 - 0.595i)T \) |
| 37 | \( 1 + (-0.746 - 0.665i)T \) |
| 41 | \( 1 + (-0.682 - 0.730i)T \) |
| 43 | \( 1 + (0.648 - 0.761i)T \) |
| 47 | \( 1 + (-0.983 - 0.181i)T \) |
| 53 | \( 1 + (-0.247 - 0.968i)T \) |
| 59 | \( 1 + (-0.247 + 0.968i)T \) |
| 61 | \( 1 + (-0.974 + 0.225i)T \) |
| 67 | \( 1 + (0.917 - 0.398i)T \) |
| 71 | \( 1 + (-0.990 + 0.136i)T \) |
| 73 | \( 1 + (-0.247 + 0.968i)T \) |
| 79 | \( 1 + (0.247 - 0.968i)T \) |
| 83 | \( 1 + (0.291 - 0.956i)T \) |
| 89 | \( 1 + (-0.113 + 0.993i)T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.4945908065957768279434743669, −21.04135747193921477647856153053, −20.02045700846650454935083062278, −19.217179141488569962499664075516, −18.72686387020291521891110039267, −18.19621940518694958918817643047, −17.24402563071486481363144610170, −15.92967646315618634716300058804, −14.931674810983623826591552119746, −14.05862085009678393502640235323, −13.83293505962217739755472217002, −12.936119438638823184697515233045, −12.022751221700137333174110453, −11.53561192455656357671989360615, −10.13191642706829737188217795539, −9.556498020864519929672491160255, −8.956687469763793153477844483911, −7.930399762718912199559352658672, −6.456183394679958909695691608571, −6.1213211705202058365982077748, −4.95521980652105063820158999937, −3.57607227962517659460373033021, −2.85792927084776203197402629137, −2.206720496750904946565206515527, −1.320433504283708336814028705633,
0.1677851963267164090561553994, 1.913687412482940582839886679000, 3.0145729000641503608675075803, 4.08186839567593680309865734817, 4.70374474263637505160207681127, 5.56887196343965593354610573885, 6.71085052661001418778811384343, 7.42574744684562519403424942129, 8.40445450059608578651957279128, 9.21185614171141519365911653778, 9.92603051340473010195655500804, 10.4995264885104752411946839768, 12.3095113414827368061690610693, 12.98902467289324729612409628931, 13.61289707109918872416834514010, 14.34785140737656565550369393516, 15.08546246349111933704589885852, 15.82751366567060013815970870611, 16.65102939985006051607363299498, 17.36303549108957587108795937960, 17.83513364604640292221774732710, 19.29020603111793433869686843719, 20.1206986617126750387605679041, 20.62112588953439971427657197870, 21.67667195563906091948629985334